Advertisements
Advertisements
प्रश्न
If `cos^-1( x/a) +cos^-1 (y/b)=alpha` , prove that `x^2/a^2-2(xy)/(ab) cos alpha +y^2/b^2=sin^2alpha`
Advertisements
उत्तर
`cos^-1( x/a) +cos^-1 (y/b)=alpha`
`cos^-1(x/a) =alpha-cos^-1 (y/b)`
`=>cos {cos^-1 (x/a)}=cos{alpha-cos^-1 (y/b)}`
`=>x/a=cos alpha cos{cos^-1 (y/b)}+sinalpha sin{cos^-1(y/b)}`
`=>x/a=y/b cos alpha+sin alpha sqrt(1-(y/b)^2)`
`=>x/a - y/b cos alpha =sin alpha sqrt(1-(y/b)^2)`
Squaring both sides, we get
`(x/a - y/b cos alpha)^2={sin alpha sqrt(1-(y/b)^2)}^2`
`(x/a)^2+(y/b)^2cos^2 alpha- (2xy)/(ab) cos alpha=sin^2 alpha- sin^2 alpha(y/b)^2`
`(x/a)^2+(y/b)^2cos^2 alpha+sin^2 alpha(y/b)^2-(2xy)/(ab) cos alpha=sin^2 alpha`
`(x/a)^2+(y/b)^2(cos^2 alpha+sin^2 alpha)-(2xy)/(ab) cos alpha=sin^2 alpha`
`=>(x/a)^2-(2xy)/(ab) cos alpha+(y/b)^2=sin^2 alpha`
APPEARS IN
संबंधित प्रश्न
Solve for x:
`2tan^(-1)(cosx)=tan^(-1)(2"cosec" x)`
Evaluate the following:
`cos^-1{cos (5pi)/4}`
Evaluate the following:
`cos^-1(cos3)`
Evaluate the following:
`\text(cosec)^-1(\text{cosec} pi/4)`
Evaluate the following:
`cot^-1(cot pi/3)`
Evaluate the following:
`cot^-1(cot (4pi)/3)`
Evaluate the following:
`cot^-1(cot (19pi)/6)`
Evaluate the following:
`cot^-1{cot (-(8pi)/3)}`
Evaluate the following:
`cot^-1{cot ((21pi)/4)}`
Evaluate the following:
`tan(cos^-1 8/17)`
Evaluate: `sin{cos^-1(-3/5)+cot^-1(-5/12)}`
If `(sin^-1x)^2+(cos^-1x)^2=(17pi^2)/36,` Find x
`tan^-1x+2cot^-1x=(2x)/3`
Solve the following equation for x:
tan−1(x −1) + tan−1x tan−1(x + 1) = tan−13x
Solve the following equation for x:
`tan^-1((1-x)/(1+x))-1/2 tan^-1x` = 0, where x > 0
Solve the following:
`sin^-1x+sin^-1 2x=pi/3`
If `cos^-1 x/2+cos^-1 y/3=alpha,` then prove that `9x^2-12xy cosa+4y^2=36sin^2a.`
Evaluate the following:
`tan{2tan^-1 1/5-pi/4}`
Write the value of sin−1
\[\left( \sin( -{600}°) \right)\].
Write the value of cos\[\left( 2 \sin^{- 1} \frac{1}{3} \right)\]
If \[\sin^{- 1} \left( \frac{1}{3} \right) + \cos^{- 1} x = \frac{\pi}{2},\] then find x.
If 4 sin−1 x + cos−1 x = π, then what is the value of x?
If x < 0, y < 0 such that xy = 1, then write the value of tan−1 x + tan−1 y.
Find the value of \[\cos^{- 1} \left( \cos\frac{13\pi}{6} \right)\]
The number of solutions of the equation \[\tan^{- 1} 2x + \tan^{- 1} 3x = \frac{\pi}{4}\] is
The number of real solutions of the equation \[\sqrt{1 + \cos 2x} = \sqrt{2} \sin^{- 1} (\sin x), - \pi \leq x \leq \pi\]
\[\cot\left( \frac{\pi}{4} - 2 \cot^{- 1} 3 \right) =\]
The value of \[\tan\left( \cos^{- 1} \frac{3}{5} + \tan^{- 1} \frac{1}{4} \right)\]
If y = sin (sin x), prove that \[\frac{d^2 y}{d x^2} + \tan x \frac{dy}{dx} + y \cos^2 x = 0 .\]
